Fujiki class C explained

In algebraic geometry, a complex manifold is called Fujiki class

l{C} if it is bimeromorphic to a compact Kähler manifold. This notion was defined by Akira Fujiki.^[1]

Properties

Let M be a compact manifold of Fujiki class

l{C} , and X\subset M its complex subvariety. Then Xis also in Fujiki class l{C} (^[2] Lemma 4.6). Moreover, the Douady space of X (that is, the moduli of deformations of a subvariety X\subset M , M fixed) is compact and in Fujiki class l{C} .^[3]

Fujiki class

l{C} manifolds are examples of compact complex manifolds which are not necessarily Kähler, but for which the \partial \bar \partial -lemma holds.^[4]

Conjectures

J.-P. Demailly and M. Pǎun haveshown that a manifold is in Fujiki class

l{C} if and onlyif it supports a Kähler current.^[5] They also conjectured that a manifold M is in Fujiki class l{C} if it admits a nef current which is big, that is, satisfies\int _{M} \omega ^{{dim _{C} M} }>0.

For a cohomology class

[\omega]\in H ^{2(M)} which is rational, this statement is known: by Grauert-Riemenschneider conjecture, a holomorphic line bundle L with first Chern classc _{1(L)=[\omega]}

nef and big has maximal Kodaira dimension, hence the corresponding rational map to

{ P} H ^{0(L} ^{N)}

is generically finite onto its image, which is algebraic, and therefore Kähler.

Fujiki^[6] and Ueno^[7] asked whether the property

l{C} is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and Claude LeBrun ^[8]

References

1. On Automorphism Groups of Compact Kähler Manifolds . Inventiones Mathematicae . 1978 . 44 . 225–258 . Fujiki . Akira . 3 . 10.1007/BF01403162 . 1978InMat..44..225F. 481142.
2. 10.2977/PRIMS/1195189279 . Closedness of the Douady spaces of compact Kähler spaces . 1978 . Fujiki . Akira . Publications of the Research Institute for Mathematical Sciences . 14 . 1–52 . 486648. free .
3. 10.1017/S002776300001970X . On the douady space of a compact complex space in the category l{C} . . 1982 . Fujiki . Akira . Nagoya Mathematical Journal . 85 . 189–211. 759679. free .
4. 10.1007/s00222-012-0406-3 . On the \partial \bar \partial -Lemma and Bott-Chern cohomology . 2013 . Angella . Daniele . Tomassini . Adriano . Inventiones Mathematicae . 192 . 71–81 . 253747048 .
5. [Jean-Pierre Demailly|Demailly, Jean-Pierre]
6. 10.2977/PRIMS/1195182983 . On a Compact Complex Manifold in l{C} without Holomorphic 2-Forms . 1983 . Fujiki . Akira . Publications of the Research Institute for Mathematical Sciences . 19 . 193–202. 700948. free .
7. K. Ueno, ed., "Open Problems," Classification of Algebraic and Analytic Manifolds, Birkhaser, 1983.
8. [Claude LeBrun]